When I study languages, my mind latches onto similarities, even when they're veiled or indirect, of form or of vocabulary. For example, I will never be able to see the word pecuniary and forget that the first syllable is related (albeit 2000 years distantly) to the English word fee, and that while both mean money now, they both meant cattle 2 millennia ago.
That said, I've begun a little experiment. I contend that there is enough of the "feel" of the Germanic linguistic tradition left in English, that you can usually find something in modern English to make aspects of old Germanic languages feel familiar. For example, last week while I was home with my foot up after surgery, I started writing down how I would explain each of the seven classes of strong verbs if I were writing an "Old English for Dummies" book. For each class there are modern English verbs that exhibit the exact same series of vowels across their principle parts--either directly, or once you account for an Old-to-Modern vowel shift, which also can be shown to be an instinctual, home-grown English thing to do.
Obviously there won't be a familiar ring to some aspects of the language--after all, there are things that make it different from its modern form (though less and less, it seems, the more I study it). For example, OE words that never made it much past the Norman Invasion won't seem familiar, although words that at least made it into Shakespeare have a chance of helping out here. Nevertheless, since we generally gain new knowledge by building on, comparing with, and filtering it through existing knowledge, it might be an easier point of access for the average Joe if different points of OE were introduced in terms of what he already knows. The closest I've seen is that many people introduce the language as a whole with examples of similarity, but then dive into the grammar like any other language, not really stopping to point out convergences along the way.
Some examples are in order.
From Old to Modern English, there was a general tendency for [a] to become [o]. We experience the relation between these sounds today. Listen to the two o's in "October". In some American dialects at least, the first 'o' is more like the a in 'farm', while the second is like the o in 'code'. (I've encountered this relationship everywhere from Russian to Hebrew, though I wouldn't include that in my grammar.)
The variation even occurs with the same letter, across dialects: I grew up pronouncing 'progress' with that short ah-sounding o, while many Americans give it a more closed 'oh' sound. So if you see a modern word like 'no', imagine pronouncing it like 'nah'. This should at least familiar. If you've done any bit of travelling, or met people from around the country, you've encountered people that pronounce words like this.
Now then, what if I tell you to pronounce the following words with that same St. Louis-sounding accent--with that real open 'ah' sound? (Like your father-in-law, gaetanus!)
They would probably sound very much like their Old English forms:
Then once you get used to that, it'll make sense to see how who came from hwa and soul came from sawol. It's just a matter of combining two things you're familiar with but aren't used to thinking about together. Like when two of your friends from different circles meet and start to date. Or like when they first invented Reese's peanut butter cups.
One more example: I could tell you that the OE strong verb Class III is conjugated like bindan: bindan--band--bunden. Or I could just get you to think about the verb drink--drank--drunk, and tell you to go conjugate swindan, sinnan, onginnan, winnan like it. Don't have the feel for this verb group yet? Recall how you instinctively change vowels for verbs like sing, sink, begin, ring, spin, sting, swim. They all have i--a--u, but if you're having trouble memorizing naked vowel sequences, then by all means, associate the verbs you learn with verbal clothing you're already used to putting on.
Is it just me? Has someone already done this and I missed it? I know I like to study from boring grammar books, but don't we want to make it easier for more people to learn this stuff--people unlike me?