Using the Karl G. Jansky Very Large Array (VLA), we have detected absorption lines due to carbon-monoxide, CO(J=0-1), and the cyano radical, CN(N=0-1), associated with radio galaxy B2 0902+34 at redshift z=3.4. The detection of millimeter-band absorption observed 1.5 Gyr after the Big Bang facilitates studying molecular clouds down to gas masses inaccessible to emission-line observations. The CO absorption in B2 0902+34 has a peak optical depth of $\tau$$\ge$ 8.6% and consists of two components, one of which has the same redshift as previously detected 21-cm absorption of neutral hydrogen (HI) gas. Each CO component traces an integrated H$_2$ column density of N(H2) $\ge$ 3x10$^{20}$ cm$^{-2}$. CN absorption is detected for both CO components, as well as for a blueshifted component not detected in CO, with CO/CN line ratios ranging from $\le$0.4 to 2.4. We discuss the scenario that the absorption components originate from collections of small and dense molecular clouds that are embedded in a region with more diffuse gas and high turbulence, possibly within the influence of the central Active Galactic Nucleus or starburst region. The degree of reddening in B2 0902+34, with a rest-frame color B-K ~ 4.2, is lower than the very red colors (B-K > 6) found among other known redshifted CO absorption systems at z<1. Nevertheless, when including also the many non-detections from the literature, a potential correlation between the absorption-line strength and B-K color is evident, giving weight to the argument that the red colors of CO absorbers are due to a high dust content.