Linear interpolation for scattered three-dimensional data.

I have a data set of scattered (non-regular) 3-D points that carry some variable and I'm trying to interpolate that variable to a new point. Currently, I have implemented a couple of methods, but I do not like the behavior of the inverse distance and I am overcoming the problems with the polyharmonic spline RHF. Is there a method that would simply give me a linear interpolation? If possible, a detailed explanation / example of a matrix configuration would be greatly appreciated.

For example (and for those who are familiar), the Tecplot software has an option for linear interpolation, but I can not find any documentation about its method.