We report important fundamental advancements in multiwall carbon nanotube (MWCNT) rectenna devices, which have potential applications for infrared and terahertz sensing and energy harvesting. Our rectenna devices incorporate ultrafast metal-insulator-metal tunneling diodes fabricated on the tips of MWCNT antennas arrays to convert high-frequency electromagnetic waves to D.C. electricity. The tunneling barrier geometry and insulator material selection are critical to designing high performance MWCNT diodes. We demonstrate new MWCNT diode structures that incorporate multiple insulating layers to control electron tunneling and achieve excellent current-voltage asymmetry (>20) with air stable metal contacts (Al and Ag). For improving antenna-diode impedance matching, we demonstrate that reducing insulator thickness and etching open the MWCNT tips can lower resistance. Our double-insulator devices show the first evidence of air stable rectification under IR and visible laser illumination (Figure 1). Open-circuit voltage is several orders of magnitude higher than previous reports that utilized single-insulator MWCNT diodes, thereby making our double-insulator MWCNT rectenna devices especially attractive for infrared photodetectors. These results bring to light a new generation of MWCNT rectenna arrays that can use a variety of top electrode materials – including air stable, transparent, and conductive materials – indicating strong potential and efficient utilization in a variety of high-frequency applications.