Actually, the correct spelling is "teaghlach", with "gh" rather than a simple "g" (you got the spelling right in your previous Notebook entry). That "h" makes all the difference.
Just a quick aside before I answer your question: technically speaking, "ea" in Irish isn't a diphthong, as diphthong refers to two adjacent vowel <em>sounds</em> in the same syllable. What you have here is a digraph, i.e. two <em>letters</em> used to represent a single sound, namely /a/. Here the written "e" serves to show that the preceding consonant is slender; it's not pronounced as such, but without it the consonant would be next to an "a" and therefore be broad.
In Irish it's very important to distinguish between normal consonants and their lenited counterparts, which are written with an "h" after them: bh, ch, dh, fh, etc. The lenited consonants have a weaker pronounciation, the air being allowed to flow through the mouth, with audible friction, rather than being blocked and then released by the tongue or lips. For example, to pronounce "gh", prepare to articulate a /g/ sound, but instead of closing off the airstream with your tongue, open the passage very slightly to let the air pass through.
Now, the lenited consonants "dh", "gh", "bh" and "mh" have pronunciations which are very close to being vowel-like -- the flow of air through the mouth is obstructed very little as they are articulated. For this reason, when they occur after or between vowels they very often combine with the vowel to create a diphthong. So /a/ + gh sounds like /ai/, as you correctly heard on teanglann.ie. (/a/ + dh has the same pronunciation; similarly /a/ + bh/mh sounds like /au/, listen to "cabhlach", for example).