I've heard that legally speaking, the strategic bombing of cities in WWII wasn't a war crime. Is this true?
In a nutshell, no. You will note that in the Nuremberg trials, no German was charged for the bombings of England.
As a general principle of law, something is legal unless there is a law prohibiting it. Although there had been some initial moves towards regulating the use of aircraft, nothing had been agreed upon by the onset of the war. As a result, the only regulation in force was the general laws of war from Hague and Geneva, which made undefended civilian targets illegal. Such targets did not exist. The targets were viewed as industrial areas used for wartime production or distribution defended by AAA. They just happened to be in cities