I did for 2 years with my first, 15 months with my second (when I decided I just couldn't keep up with the laundry anymore). A word of warning, cloth diapers can be very expensive
As justbishop said there is a whole new world of cloth diapers out there. However, if you are going budget you are probably going to be sticking to more budget friendly options, and opting out of all the new and fancy stuff.
For budget diapers flats or prefolds, and a cover with snaps or velcro are going to be the way to go. Check out Thirsties covers
http://www.amazon.com/dp/B001TUYHCC/?tag=yahhyd-20&hvadid=62826779011&ref=pd_sl_8599khher9_b
they are reasonably priced, work well, and hold up well to washing.
I found I didn't like prefolds at all. The next step up is going to be a fited diaper. A fitted diaper has velcro or snap closure, but still requires a water proof cover. I actually sewed up a bunch of my own, and saved a good deal of money. If you sew its a good way to go. You can buy materials specifically made for diapering (bamboo fleece, banboo velour) or even use stuff you have on hand (old towels, fleece shirts, cotton tees, cotton sheets). I have also heard great things about Crickets diapers
http://www.crickettsdiapers.com/diapers.html
The next step up and the cadilac of diapers are going to be your pocket and all in one diapers. Which is exactly what the name says. Basicly a cloth version of disposables. My absolute favorite dipe was sposoeasy
http://www.greenmountaindiapers.com/allinone.htm (and also the store in this link is an awesome place to start!). The sposoeasy is as easy as it gets. I liked them in velcro because I got a better fit.
Another option is buying used to save money. You can go to diaperswappers.com and find tons of information, and lots of used dipes for sale. But 2 things to beware of. It can be a place with tons of "mom wars" and lots of threads about "i'm a better parent because..." So stay away from the off topic and parenting board. And also be very wary of who you are buying from. Make sure they have a lot of feedback, and read their feedback. There have been many many many cases of mom's getting scammed over there, and its not very well controlled.
As far as the gross factor, not so much as you would think. When they are little the poo just washed right off in the washer. When they get a little older, you can buy a disposable liner for the dipes when they poo you just dump it with the liner into the toilet. Washing is easy. Store dipes in a wetbeg, most will fit right into an average size garbage pail. Plan on washing every other day or you will get odors. My wash routine was cold rinse with no detergent, hot wash with some tide (I tried many detergents, tide was the only one that worked for me), cold rinse, and extra cold rinse. Dry in the dryer.
Some tips, don't expect to go right from the start. Plan on starting once the babe is a couple weeks old. Those first few weeks are so rough, you will be setting yourself up for failure if you expect too much. Don't buy a ton of one kind of diaper to start with. Start by buying a couple each of a couple differant diapers, try them out then choose what you like and get some more. And remember it doesn't have to be all or nothing. No reason you cant alternate between disposables and cloth, you would still save some money. Good luck with the new baby!