The cheapest would be by bus but it's obviously not the quickest !
From Chiang Mai you could head east and visit the villages along the Mekong in the way to Cambodia but the southeastern part of Thailand is not very nice.
A great path if you have time would be to cross the Lao border for the 2 days trip by slow boat to Luang Prabang and go south to cross the Cambodian border in the 4000 islands. Besides the stays in each stop, you are looking at : 2 days boat for LP, 1 day bus for Vientiane, 1 day for Pakse, 1 day for the 4000 islands and 2 days to reach Siem Reap so it takes time but you get to visit wonderful Laos on the way.
But if you don't have the time to spend time in little places on the way (like Vang Vieng, the Bolovens, Done Kong in Laos and Banlung or Kratie in Cambodia) it's just not worth it.
I know 2 GH in Siem Reap if you need it :
Basic rooms from US$ 3 at the happyangkorguesthouse.com
Nice comfortable place from US$ 19 at the secretguesthouse.com
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