Plants can be considered automatons: for instance, they bend towards light because light causes release of plant hormones, which cause the plant to grow asymmetrically (light side of the stem grows faster or slower than the other, causing "bending"). This is a gross oversimplification, but the basic idea is that plant hormones control the plant, there exists inputs to the system and outputs are the excretion of plant hormones. It sounds like a computing device, so could such a system be programmed?
Now, suppose it would be possible to take a simple plant, and "program" it somehow to behave in a certain way as it grows. For example, one could grow a simple trellis out of bamboo. The trellis would act as a building block of a "skeleton" of some structure, e.g. a house. By connecting many such bamboo trellises together like Lego blocks one could create a bio-degradable, lightweight, yet relatively rigid structure.
Other uses for "programmable plants" could include moss growing in certain aesthetic patterns such as fractals, geometric figures, something out of Lindenmeyer systems, mosaic-like things, or any figure sufficiently simply explained as a short algorithm, properly encoded into the genome of the plant.
How could this be done at all, in theory? One possibility would be through "time-release" plant hormones: signals to turn the plant left would be triggered once the plant has spent the time to grow 10 cm, then to left again after the plant has grown another 10 cm. The result would be an upside-down letter U.
Notice the close similarities of the growth mechanism and turning (or otherwise altering the growing "direction") to Lindenmeyer systems. With relatively simple operations it is possible to create complex geometrical shapes.
Time-release plant hormones are an example of signals directing the plant "internally". The plant could also be directed "externally" through the environment. For example, hormones would be activated within the plant by different kind of environmental conditions: lights of different colours, vibration, temperature, pressure, angle of light source, etc. By altering in certain patterns the attributes of the environment which cause an internal response within the plant, one could program the plant to grow in certain spatial structures.
Update 071206: Even though it is with non-genetic means and a lot of patience, it's worth mentioning that some people are growing furniture.
 Prusinkiewicz, Przemyslaw; Lindenmeyer, Aristid: The Algorithmic Beauty of Plants. Download PDF version.