Synthesis of methylenecyclohexane from cyclohexylmethanol
Whichever method of elimination you end up preferring, there is no way to directly eliminate the hydroxy group without derivatising it first in some way. The hydroxide ion is a very bad leaving group (but a rather good nucleophile) and transforming a hydroxy group into a better leaving group by protonation ($\ce{R-OH2+}$) requires acidic conditions that do not really allow the (basic) abstraction of a &#x3b2; hydrogen. For each of the simple options:


A hydroxide ion will never leave as part of an $\mathrm{S_N2}$ or $\mathrm{E2}$ reaction; it is too weak a leaving group.
An $\mathrm{E1}$ reaction would require strongly acidic conditions to protonate the hydroxy group; under these conditions the &#x3b2; hydrogen cannot be abstracted. $\mathrm{S_N1}$ would dominate.
An $\mathrm{E1cb}$ mechanism is absolutely impossible. The $\mathrm pK_\mathrm{a}$ value of the &#x3b2; hydrogen is around $50$ while that of the hydroxy group is around $15$. The only conjugate base being formed will be the alcoholate anion.


This means we have to resort to two-step procedures. A number of them have been published in literature in various different contexts. Some bear a name, others don’t. Some follow one of the paradigms mentioned above, others follow a completely different mechanism. Since there is only one elimination product possible, you need not worry about selectivity. I shall present a few possibilities grouped by the mechanistic type where possible but note that this list is not exhaustive and does not even begin to scratch the surface.

$\mathrm{E2}$ type mechanisms

The simplest is the one Orthocresol mentioned in the first comment: transform the alcohol into the corresponding tosylate, e.g. by adding $\ce{TsCl}$ and pyridine, then eliminate with additional potentially stronger base.

A variant of this would be to use mesyl chloride/DBU instead. Depending on how activated your proton is, a simple addition of more DBU to the reaction mixture may suffice.

Considering that your starting material is a primary alcohol, $\mathrm{E1}$-based reactions are very unlikely. Considering again the acidity of the &#x3b2; hydrogen, so are $\mathrm{E1cB}$; no matter which the reaction protocol.

syn-selective eliminations based on cyclic transition states

One example is the ester pyrolysis: use an acid chloride (e.g. acetyl chloride $\ce{AcCl}$) to synthesise an ester, then heat up to initiate thermal rearrangement to the corresponding acid and alkene. The transition state is cyclic, so this reaction is syn-selective.

A similar reaction is the Chugaev elimination. React the alcohol with a base and $\ce{CS2}$, then add an alkyl halide (typically iodomethane) to generate a xantogenate. This also fragmentates thermally to give an alkene and a dithiocarbonate while syn-selectively removing the &#x3b2; hydrogen.

And finally my favourite one: the selenoxide pyrolysis. First synthesise a selenoether with $\ce{NCSe-C6H4-NO2}$ (ortho-nitrophenyl selenocyanate) and $\ce{PMe3}$ in a Mitsunobu reaction, then oxidise the selenoether e.g. by using $\ce{H2O2}$ at room temperature to smoothly give the corresponding alkene — again via a cyclic transition state practically identical to the former two.