This has been discussed many times. See Seat 61. Get to Pakse and go by bus to Vientiane then up to Luang Prabang and bus or slow boat to Huay Xai. Then bus to Chiang Khong, and on to Chiang Mai. I'd say about 3 to 4 days travel time without stops-someone else may know exactly. Or, fly to Chiang Mai for $125 on AirAsia- which is what I would do.

If you're going to make a move anywhere around Songkran, do it early and plan to ride out the long weekend (Sat-Sun-Mon, 13-14-15). Buses and borders do not stop for the festival, but it gets busy. Try to get a "Green Bus" to Chiang Khong. Or spend a night in Chiang Rai, see the white "temple" then get local 2.5-hour bus to Chiang Khong.
http://www.greenbusthailand.com/website/en/services/bus/time-table
Cross the bridge to Huay Xai, then get bus (12+hrs) or slow boat (2 days) to Luang Prabang (refer above hobomaps link). If arriving LP in the middle of Songkran (Pee Mai Lao) expect to get soaked arriving in the city, and accommodation will be quite full, so maybe plan it out, pre-book a room. That, or just stay in CM until it's over.

As above. 3 weeks is nothing to see SE Asia. Way too short a time for all of 3 countries.

But if flying into Hanoi and out of Bangkok, a good itinerary might be:

A week for exploring north Vietnam
Fly to Siem Reap from Hanoi
4 days for Angkor Wat/Tonle Sap/Siem reap
fly to Bangkok, 3 days there
Trip to the southern Thai islands and back to Bangkok with the rest of your time

Or you could skip Cambodia,

and fly to luang Prabang, Laos instead
3 days in Luang Prabang
2 day boat trip on the Mekong river from Luang Prabang to Huay Xai
Huay Xia by bus to Chiang Mai, Thailand
3 days in Chiang Mai, plus 2-3 day jungle trekking trip from Chiang Mai
overnight train to Bangkok

That would be another good way to spend 3 weeks in SE Asia.

My reply is probably coming too late (sorry for that, I don't come on this forum very often, turns out today I needed some advice myself for an upcoming trip to Myanmar) but here's my suggestion (this may be useful for others thinking about doing the same).

From say Siem Reap to Chiang Mai by road is very easy - via Thailand. You catch one of the international buses to Bangkok and then another bus or train to Chiang Mai. Easy peasy. Or if you are more adventurous, you can catch a bus to O'Smach, then find a songthaew to Surin and from there you can easily find one of many buses to Chiang Mai, either changing buses in Udon Thani or somewhere or there might even be a direct option (I know there's a Ubon Ratchathani-Chiang Mai bus, which may or may not pass through Surin).

I wouldn't go via Laos unless you really want to see that country. Otherwise, you are going to be wasting time and getting a sore behind from sitting on uncomfortable buses on some of the region's worst roads. The road from the Cambodian border to Vientiane is decent enough and relatively flat and straight, but will still take ages. From Vientiane to Luang Prabang is a roller coaster and then from Luang Prabang to Huay Xai you have the boat (not sure if it runs during the rainy season?) or an excruciating bus ride. Then you change to the Huay Xai (Bokeo)-Chiang Rai international bus, then Chiang Mai. OR you take the once daily 18 hour direct international bus from Luang Prabang to Chiang Mai. It's operated in tandem by a Thai or Lao bus on alternate days and is much more comfortable than ordinary Lao buses and much easier to stick with one bus than changing multiple times BUT it's still a long haul.

An alternative is ending your Lao adventure in Vientiane, then catching an international bus to Udon Thani, where you catch a cheap flight on Nok Air or Air Asia to Chiang Mai.

Chiang Mai and or Luang Prabang would suit. Dont have time for extensive exploring.

Are you planning on being in Chiang Mai for the festival?

I thought about doing the 2 day trip to Thailand from LP instead of flying to Chiang Mai, but 2 days is longer than I'd like to spend (unless people really think this is not to be missed).

Well really it would take 3 days out of your trip, because you'd need most of a day to travel from Huay Xai to Chiang Mai. So probably don't.

Could we have a day like this near LP, or would it be better to go to NK?

Not sure exactly what road(s) these tourists used on their bike trip. I can't see why the country roads west of Nong Kiaw would be more scenic than the roads south/east of LP. If you want to see the Kuang Si waterfall, you could rent a bike that day, and use that, then after the waterfall continue south along the Mekong river, and you can make a loop trip back to LP on a different road.

The trip from Nong Khiaw to LP would have been perfect, but I understand it isn't possible due to dam construction. Is that still true?

Yes. It's about 4 hours by minibus on the road these days. Nong Kiaw is interesting for the nearby caves, and the epic views you can get from the viewpoint you reach hiking uphill from the town. And also the boat trips on the Nam Ou river to Muang Noi too.

Basically I think if you want to do Luang Prabang plus Nong Kiaw too, then you'd spend only 3 days in Chiang Mai. I like your idea of including a day for Vientienne: that makes it more of an authentic visit to Laos.

And yes, if you want to go motorbike exploring in the countryside, much better from LP in Laos than from Chiang Mai.