I've seen a number of techniques for laying individual elements at 45° angles, but I need a consistent method for laying down larger plates at 45 degrees while maintaining close contact with the base plate layer below it.
What techniques are available to achieve this?
45° angles are tricky because of the dimensions of 45-45-90 triangles:
The fact that the hypotenuse needs to be a multiple of radical two makes it difficult to build out of LEGO plates which are generally limited to integers or halves. The best you can do in a reasonably small amount of space is a 5 x 5 x 7.07 triangle, but that doesn't come close enough to work well.
I'm not aware of a way to do this using just plates, but if you are willing to use other bricks as well, there are some ways. For example, some older bricks and most transparent bricks don't include tubes within the bricks. This allows constructions that are not limited by whole numbers:
Edit: There is now a specialized 45° angle plate (15706) that can be used for this purpose:
Hinges can work, I have used this technique before, if you sandwich the hinges between two plates you get 45 degree 'bricks' that are quite solid.
In most cases, tiles and plates in the right place will keep sufficient contact with the base plate to have a stable model. 
It appears that some LEGO pieces are designed with the proportions of 45-degree right triangles in mind. For example, the 3x3 plate with one corner removed has a diagonal edge that is very close to three studs long, as can be seen in this construction:
Likewise, the 8x8 corner plate has a diagonal edge that is very close to 10 studs long:
This is in old question but I think many people find this helpful. I accidentally found this Lego technique where you put a 4x4 round plate between four 1x1 round plates. You can see in the picture how to do it exactly. I think it is an "illegal" Lego technique. If you look closely you can notice that the distance between each 1x1 plate is a tiny bit more than four studs. And the cutouts on the side of the 4x4 round plate are designed for the normal four studs apart in my opinion.
The turntable is just there so it doesn't come apart, but the 4x4 plate is fully locked by the four 1x1 plates themselves. It may look like it doesn't work but just try it out.
If near enough is good enough you could build based on a 3:4:5 right angled triangle: i.e. put "dots" on the base plate 3 up and 4 across from each other which would then connect to the circles underneath a baseplate 5 holes apart.
The angle would then be ~37 degrees or ~53 degrees between the two structures.
Possibly I’m the same Woody that previously answered this question. If so, I didn’t describe it well 5 years ago.
This is a cut down version of the 7/7/10 triangle which is slightly better than the 5/5/7 triangle.
Placing dots (1x1 round plates) on the bottom of one plate with 5 spaces in between will line up perfectly on a lower plate 4.5 down and 4.5 across (one dot fitting between studs on lower plate)
