{\sl Bidimensionality Theory} was introduced by [{\sc E.~D. Demaine, F.~V. Fomin, M.~Hajiaghayi, and D.~M. Thilikos}. {\em Subexponential parameterized algorithms on graphs of bounded genus and {$H$}-minor-free graphs}, J. ACM, 52 (2005), pp.~866--893] as a tool to obtain {\sl sub-exponential} time parameterized algorithms on $H$-minor-free graphs. In [{\sc E.~D. Demaine and M.~Hajiaghayi}, {\em Bidimensionality: new connections between {FPT} algorithms and {PTAS}s}, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2005, pp.~590--601] this theory was extended in order to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this work, we establish a third meta-algorithmic direction for bidimensionality theory by relating it to the existence of { linear kernels} for parameterized problems. In particular, we prove that every minor (respectively contraction) bidimensional problem that satisfies a separation property and is expressible in Countable Monadic Second Order Logic (CMSO), admits a linear kernel for classes of graphs that exclude a fixed graph (respectively an apex graph) $H$ as a minor. Our results imply that a multitude of bidimensional problems admit linear kernels on the corresponding graph classes. For most of these problems no polynomial kernels on $H$-minor-free graphs were known prior to our work.