Give me your time frame: when you said "for the month of June" I wasn't clear whether the whole month or just during June a couple of days.
If will you be staying a month in Europe, I'd suggest using a train pass (compare both the passes bought from the USA and the ones bought in Europe).
Will you do the backpacking way or more city-trip style?
Elaborate a little please, I've traveled both ways: From a 1 month trip starting in Belgium visiting with the train Italy, Slovenia, Greece, Turkey, to end up in Israel, having spend around $950 in total, all inclusive. (yes its possible: we left a trail of depressed or suicidal people behind us, as an immediate result of our soft, considerate and diplomatic bargaining....) to First/business trips with stopovers and stays in decent hotels.