I ran into this same problem. Just ordered and received the BenQ G Series LCD monitor 27" and it did not come on even when connected via HDMI. I fiddled with it for a minute and found that there are switches on the monitor. If you are connected via HDMI you have to select that.
I don't know how other models are set up, but the switches on mine are right by the power switch. You can sorta follow your nose through the process.
As soon as I did that, it came right up.
The monitor looks pretty decent. I have it sitting beside my fairly new Apple Thunderbolt display. I was pretty worried that by comparison it would be terrible, but it's not bad at all. I just couldn't afford two Thunderbolts.