Traveling from Thailand to Laos to Vietnam

The short answer is 'yes', but whether you will depends a lot on your modes of transport, river travel is slow and not so frequent, and how long you are hoping to stay in each destination. Southern Laos is slow to travel through, Thailand and Vietnam are a breeze. But, 3.5 weeks is most definitely doable for going through all three countries.

However, it appears you will have to get back from Vietnam to Thailand. You should strongly consider doing that by budget flight.

It's really not very hard to get a sense of where you can travel, taking how long, by simply buying the Lonely Planet for the region and planning your trip.

I will add to MastaBaba's very relevant answer by suggesting a specific route as per your request. This is based on personal experience (all of this, not in this order, some of it in the opposite direction) between March and May 2014.

If you start in Bangkok, you can go all the way to Vietnam by bus while visiting the places you mentioned in Laos, and then go down to Ho Chi Minh City by train. You can get your Laos visa on arrival at the border and your Vietnam visa at the consulate in Luang Prabang. Check visa requirements beforehand; you will need to have pictures with you for one thing.

1. Bangkok to Ayutthaya by bus.

2. Ayutthaya to Sukhothai by bus.

3. Sukhothai to Udon Thani through Phitsanulok by bus. To my knowledge, there are no direct buses from Sukhothai to Udon Thani, but it takes about an hour from Sukhothai to Phitsanulok, where you can catch a bus onward. Nothing very interesting in Udon Thani: spend the night if you have to, but don't waste time there if you have a limited amount.

4. Udon Thani to Vientiane by bus, crossing the Thailand-Laos border. This takes about an hour and a half.

5. Vientiane to Vang Vieng by bus.

6. Vang Vieng to Luang Prabang by bus. Spend a few days in Luang Prabang while waiting for your Vietnam visa. You can expedite things by paying more if needed. I actually got mine same-day.

7. Luang Prabang to Phonsavan by bus.

8. Phonsavan to Thanh Hoa through Sam Neua (also spelled Xam Neua). You will have to spend the night in Sam Neua, and probably also in Thanh Hoa. Nothing very exciting in both these cities, so you can move along. Also, Sam Neua is not a typical foreign tourist destination, which means even the hotel staff will likely not speak English, so it can be a little awkward.

9. If you have time, from Thanh Hoa to Hanoi by train. The capital is interesting to see. Also, if you can squeeze in a two-day cruise on Halong Bay, which can be arranged from Hanoi with transportation included, it will definitely be a highlight of your trip.

10. From Thanh Hoa or Hanoi down to Hue by train.

11. From Hue to Hoi An by bus.

12. And finally, from Hoi An to Ho Chi Minh City through Danang. There is no train station in Hoi An, so you will need to arrange (easy to find) transportation to nearby Danang and take the train from there.

You mentioned Nha Trang: I did not go there, but there is a train station in Nha Trang, so you can stop there on your train journey from Hoi An/Danang to Ho Chi Minh City.