What physical evidence exists that shows motor proteins "walking" within a cell?
Since you chose kinesins as an example class of motor proteins, I will also stick to them. In general, the movement mechanisms of kinesins are well studied and the general structure of kinesin proteins, that is a dimer, is known by crystallography (see Kull et al. (1996) for the structure publication). 

Even before that, the movement of single kinesin molecules in discrete steps of approx. 8 nm was shown by Svoboda et al. (1993). To resolve single molecules, they used optical trapping interferometry (see this practical guide to optical trapping by the Princeton University for more detailed information). Four years later, Schnitzer and Block (1997) were able to show that that for every step of approx. 8 nm lof a kinesin molecule, one ATP is hydrolysed. Visscher et al. (1999) used molecular force clamps to study ATP and load dependent characteristics of single kinesin molecules.

So far, it appears to be that kinesins move by a 'hand-over-hand' mechanism: Yildiz et al. (2004) labelled single heads of kinesin dimer molecules with a fluorophore and found that their data are congruent with a step of approx. 17 nm followed by a step of approx. 0 nm which indicates that the two dimer hands alternate in movement.

See also the review by Hirokawa et al. (2009) for a general overview of kinesin movement.