My husband and I do not have a huge budget, know very little about this area, and are looking to mostly be in nature environments (as opposed to big cities)

Any suggestions?

Northern Thailand might suit you...trekking, rafting, biking, climbing. The Similan Islands are reportly very stunning diving/snorkeling. Phang Nga...those amazing cliffs and caves.

In Vietnam, Halong Bay is also reportedly gorgeous. Some of the Vietnam coast looks nice to me, too, and there are some very reasonably priced seaside resorts that look nice. Sapa and surroundings is on our list of places to go.

Although you say you don't like cities, many, including myself, love it. I'd give it a couple of days. You got some good ideas for Thailand in reply to your other post.

3 weeks isn't really very long...I'd think about limiting your destinations to 4 or 5 so that you have time to relax and know a place a bit. Travel between destinations takes on average 1/2 of a day, even flying.

In Vietnam I would go to Hanoi and the north and in thailand I would focus on northern thailand -- see my off the beaten path northern thailand thread where I describe almost 4 weeks in northern thailand for ideas.

From Chiang Mai you can fly to Luang Prabang, Loas. Limited flight schedules but you might also check re: connecting into Vietnam. If you do the beach areas and caves in Vietnam you might then fly to the northern territory to enjoy country/river activities (if you don't dive).
Luang Prabang is very cost effective and has a lot of trekking/nature--like Chiang Mai 30 years ago.

Check the websites for Bangkok Air and Thai Airways as they are doing various promotions in-country and between countries. One item that will add up is the cost of visas, photos and departure taxes as you leave each destination so keep that in mind if you are really tight on $.

In Vietnam look at Sapa area - specifically the Topas Eco Lodge property looks stunning and is on my list of must dos. Hanoi-Sapa is an overnight, fairly cheap transfer by train. Also take a look at the www.ecoclub.com site for anything that appeals - we stayed at their Laos property near Vientiane and loved it.

From Hanoi to Thailand you can fly with airasia ( now they are promotig free ticket to Thailand for travelling in 2006)
3 week I think it is not long but enough time to visit main place such as Ha Long, Sapa, Hanoi, Hue Hoian ( in hoi An should go at weekend because they have lantern festival.