You are right, I was using the hand grip monitor. I had wanted to use a chest-strap monitor from the start, but there was nothing in my elliptical's documentation that said a wireless monitor was supported.
However, per your suggestion, I purchased a wireless monitor with chest strap and tried it this morning. To my surprise, my elliptical picked up the wireless signal. Also, using the chest strap my heart rate is fairly regular without pauses.