Perfect light absorption in the visible and near-infrared (NIR) was demonstrated using metamaterials, plasmonic nanostructures, and thin films. Thin film absorbers offer a simple and low-cost design as they can be produced on large areas and without lithography. Light is strongly absorbed in thin film metal-dielectric-metal (MDM) cavities at their resonance frequencies. However, a major drawback of MDM absorbers is their strong resonance iridescence, i.e., angle dependence. Here, we solve the iridescence problem by achieving angle-insensitive narrowband perfect and near-perfect light absorption. In particular, we show analytically that using a high-index dielectric in MDM cavities is sufficient to achieve angle-insensitive cavity resonance. We demonstrate experimentally angle-insensitive perfect and near-perfect absorbers in the NIR and visible regimes up to ±60°. By overcoming the iridescence problem, we open the door for practical applications of MDM absorbers at optical frequencies.