you should be able to hear beeps even if hte cpu don't work. The mobo takes care of the beeping. However check and make sure the speaker is in there right. If its not that might be a issue of y your not hearing it. Does the screen display anythign while it is on? IF it doesn't then it might end up being a ram problem however if ur psu did go bad the cpu might go bad as well.
Also you well need to but a fresh install of windows since u have new hardware (motherboard) if u didn't do that. It might simply be overheating.
If it is overheating make sure the fan and everything is working and pluged in. Also when u first start the computer up does is it on longer then shuts off then when u restart it, does it like shut off at a shorter point? If it does that then it might be overheating.
Also when the psu went out it could have fired hte psu also. There has been cases of that happening when psus go bad. However before you spend anymroe money just make sure you check every possible thing.