Please provide cheap but clean lodging in Prague in early June
Try vrbo.com Last May we rented a 3 bedroom apt through this site, and it was very affordable. Plus you get an apt, so you can shop at the stores and cook like a local. My family loved it.
Have you looked in the RS Prague Guide Book yet? It has numerous recommendations that might fit you request.
We had a great stay at Guest House Lida - it's a little out of the city center but near a metro stop. You can check out their website for rates and Trip Advisor for reviews.
http://www.hotelmetamorphis.cz/ Metamorphis was awesome for me. It is right outside city square making it perfect walking distance to just about everything.
How cheap is cheap? It differs from person to person. We stayed at Hotel Hastel at what I thought was a reasonable price. Less than 70 euros.
Look at Miss Sophies...excellent staff and aparmtent! http://www.miss-sophies.com/