No. That is not to say there was no use of trench warfare on the eastern front, just that the theater maintained a much more mobile character than the static west. If you consider the sheer size of the Eastern Theater than you can begin to understand the difficulties both the Russians and Germans would have had to establish a system as extensive as the one in France and Belgium. The reason that trench warfare was so effective on the Western Front was that it was a completely interlocking defensive network, from the coast to Switzerland – it was an enormous distance, but close enough that both sides could rush reserve divisions to reinforce any point in the line that was about to collapse, which is why all the massive offensives in the west resulted in massive casualties, but relatively little territorial gains. Replicating this same paradigm in the east would have been nearly impossible due to the sheer size of the front, the time it took to bring up reserves to different points in the line, and the communication system, which was more primitive than in the West. Trenches were utilized, but the Germans discovered in their 1915 offensives that by coordinating artillery and infantry to attack cohesively, they could overwhelm the Russian trenches and resume mobile warfare, much to the satisfaction of the OberOsts.

Sources:German Strategy and the Path to Verdun: Erich von Falkenhayn and the Development of Attrition, 1870–1916; The Eastern Front: 1914-1917

/u/GrandMarshal answered the question very nicely, but I also wanted to add in a couple of maps of the Eastern Front.

The first map highlights the situation on the Eastern Front at the beginning of hostilities in 1914. Notice that the battle-lines run throughout Poland near the border of the German Empire in Silesia and the Russian Empire which controlled large areas of present-day Poland.

The second map shows the front at the time of the Brusilov Offensive in September 1916. Notice how the the left side highlights each front.

The Western Front was basically unchanged from the First Battle of the Marne, showing how trench warfare heavily restricted mobility.

The Italian Front is similar in that the front shows the presence of trench warfare between the Italians and the Austro-Hungarians (although the lack of mobility could be attributed to the Alps causing restricted movement as well as the heavy amount of fighting on the banks of the Soca (Isonzo in Italian) River.

Looking at the Eastern Front, however, it is noticeable how much the front had shifted eastward in only two years. Whereas the front was on the German-Russian border prior to the Russian invasion of East Prussia, the Central Powers had heavily pushed the Russians back to at least Ukraine and had control over Poland and Lithuania at the time of the offensive.