Do you get any beeps when you start it up.
Either way, shutdown, turn the PSU off at the swtich on the computer and at the wall (leave plugged in). Remove the CMOS battery, hold PC power button down for 20 seconds, replace CMOS battery, turn PSU back on at computer and at the wall, try again.
Furthermore the HD6790 has a TDP of 150W (12.5A) and your PSU only can provide (max, derated) around 13.5A (18A rated). Even if it could (it cannot) provide the rated 18A on the 12V rail this is clearly insufficient. You need a new PSU either way. Best case scenario, the TDP is closer to 115W (~10A), your PSU can provide 18A, you're still cutting it very very fine as it has to power everything else on less than 9A = ~100W.
When your CPU requires up to 84W, that doesn't leave much (16W) for the motherboard, HDD, fans, RAM, etc etc.
Finally, if the PSU you have is the one i think you have, it only has 1 x PCIe connector and your graphics card requires 1 6 pin + 1 x 8pin. If this is true, you should suspected your PSU straight away as not being up to the job. Secondly, even AMD recommends a minimum of 550W.
You need a new PSU. Something like this is required.
Corsair 500W with 38A (456W) on the 12V rail for $30 after rebate.
Failing to replace your PSU may lead to loss of several components, motherboard, ram, cpu, gpu and could cost you a lot more than $30.