I'm still new to this, but what I did was put those I gave awards to in my blog roll. My logic being that if I like the blog enough to give it an award, I like it enough to check back often - hense my blog roll. So by extension, I'm now also using blog awards for a specific reasons - but so far I'm finding that works.
I also find I tend to mainly use my blog roll when I'm busy - rather than my reader - so I'm up to date on the blogs I enjoy most.
However, it also sometimes delays my posting awards I get until I have enough new blogs I want to add to my roll. So I can see this getting a bit unmanageable if I keep up the practice - but it's working for now.