First off, African-Americans are a highly diverse group and that makes it very hard to extrapolate any collective attitude towards the Soviet Union. Often, the most articulate responses to the Soviet's antiracism campaigns and slogans was from activists, and their attitudes should not be accepted at face value. Some were genuine believers in Marxist-Leninism or other radical philosophies, other black intellectuals would wield Soviet actions as club against Jim Crow to embarrass the US government (and of course, many fell between these two poles). Secondly, attitudes towards the USSR are going to vary widely between 1917-1991. The geopolitical and domestic context are going to vary widely. Although being a member of the CPUSA was never a safe position in US politics, activism was far less restricted during the Depression than the 1950s and 60s.

Although the Bolsheviks were roughly aware of American racism, it was actually black activists that brought lynching and other problems of Jim Crow to the attention of Moscow. Black radicals from the US and Caribbean brought up the issue of racism in the Americas during the 1st Congress of the Comintern in March 1919. The Fourth Comintern Congress authorized a "Negro Bureau" three years later and under its auspices published a number of tracts attacking the racism and publishing books like Negroes in America and Lynching in America by Claude McKay. These books had a limited print run in both Russian and English. McKay was a Jamaican-American poet and author whose dark skin meant that he was an ideal poster-child for the Soviet's antiracist ideology. Soviet publications in the early 1920s featured McKay with Soviet leaders.

The antiracism campaign trailed off in Soviet discourse in the early 1920s, but picked up speed again in the early 1930s. There were multiple reasons behind this. Firstly, as the Five-Year Plan gathered steam it imported American expertise and methodologies and "Americanism" became a byword for modernity for many within the Soviet Union. Emphasizing Jim Crow was a way for Moscow to temper some of this enthusiasm. Also, positioning itself as the antithesis of biological racism was a means for the USSR to set itself up as a clear rival against Nazi Germany. But is should be added that some of this renewed pressure came from activists within the Third International itself as the Depression led to renewed anti-black violence. These individuals saw an increase in racism as a symptom of the terminal stages of bourgeois capitalism. These activists, both white and black, felt that the Depression created an opportunity to mobilize among black workers and tenant farmers. Other African-Americans saw the Soviets and their affiliated labor parties as relatively color-blind and open to membership. The CPUSA and the USSR also gained a degree of fame for its leading the appeal of the "Scottsboro Boys," nine teenagers accused of two white women. The Soviets encouraged popular support for their "revolutionary brothers" through various forms of public demonstrations. Although the cases predictably ended in convictions for most of the defendants in 1932, the case reflected poorly on the US justice system, especially since the Soviets actually prosecuted and expelled two white American laborers who assaulted a black man working in a Stalingrad tractor factory in 1931. The thirties also saw an influx of blacks from the Americas into the Soviet Union, some as intellectual activists, others as laborers.

However, despite this, the USSR and African-Americans were seldom in synch in their antiracism in the 1930s. Moscow's semi-official stance on African-Americans was that they were a captive nation within the US, which was not the opinion of many black activists. The Soviet's antiracism again trailed off towards the end of the decade as it became subsumed into the wider antifascist crusade. This rankled some black activists as it meant the Soviets were sometimes more concerned with organizing blacks not against Jim Crow, but more to influence Washington's foreign policy. Barbarossa accentuated this trend as Moscow tended to be less interested in movements like the Double-V campaign and A. Philip Randolph's proposed march on Washington, which the CPUSA denounce. Black CPUSA members were often left to their own devices.

The Cold War led to yet another renewed emphasis upon antiracism in Soviet agitprop, but the McCarthyite climate of the 1950s and 60s made it much harder for African-Americans to embrace it. Here the experience of Bayard Rustin is tragically typical. Rustin, who operated under the double burden of being having joined Communist parties and being a homosexual, was a member of MLK's inner circle and was a keen political organizer. MLK often had to distance himself from Rustin though for obvious reasons as his existence lent credence to the popular canard amongst defenders of Jim Crow that the Civil Rights Movement was riven with Communists and other subversives.

So, in answer to the OP's original question, yes, African-Americans were aware of the USSR's antiracism and were in no small measure responsible for its original articulation. The Soviet's ulterior motives behind using "And your lynching negroes" was often quite apparent, but also relatively immaterial at times. That the Soviet Union had taken a stance against racism enjoyed some currency among black intellectuals in the Americas, especially during the 1930s. The harsh intellectual and political climate of the 1950s stifled any overt expressions of support from African-Americans for the Soviet attitude. By the late 1960s, the New Left and the rise of passionate, and often non-white revolutionaries in the Third World like Mao or Che meant that many black radicals gravitated towards them and not the increasingly geriatric USSR.

Again, this is only a (rather poor) surface summary of a very complex issue.

Sources

Kelley, Robin D. G. Hammer and Hoe: Alabama Communists During the Great Depression. Chapel Hill: University of North Carolina Press, 1990.

Roman, Meredith L. Opposing Jim Crow African Americans and the Soviet Indictment of U.S. Racism, 1928-1937. Lincoln: UNP - Nebraska, 2012.

Zimmerman, Andrew. Alabama in Africa: Booker T. Washington, the German Empire, and the Globalization of the New South. Princeton, NJ: Princeton University Press, 2010.

Paul Robeson states in this autobiography that he was very fond of the USSR when he visited there. He was treated much better there than here. He was planning on having his son go to college there but then Cold War tensions got a little more heated. The US government also used his USSR visits as justification to take away his passport so he couldn't sing in Europe.