There are a few different questions embedded in your post. One is about the technique involved. But the other is just about musical ideas. It is easier, I think, to learn the techniques involved in intricate flatpicking and crosspicking, but where to come up with ideas is a different matter altogether.
I don't want to function as an advertisement, but Stephen Seifert recently put a lesson on the Dulcimer School about crosspicking exercises. They would be really helpful. Each one is a pattern along the lines of what Ken H suggests. They are good to practice to get your picking hand up to speed. But of course, when you actually play a song, you will not follow a pattern but vary your picking enormously. I think if you join the Dulcimer School (it's only $15 a month) you'll find a lot there for you. In addition to that lesson on crosspicking exercises, Stephen has a lot of lessons where he teaches a song from a simple beginner version to more advanced versions. Paying attention to how you "fancify" a song will give you ideas you can import into all your playing. And Stephen even has that Black Mountain Rag available in a super-slow video so you can watch his hands really closely, although I have yet to attempt that song.
My first advice is to work on your right hand picking. You say you are satisfied with your strumming right now. I assume that means that you have developed a very steady in-out or out-in strumming pattern. If that is the case, you have the basics down. Next try to skip a beat, accent a beat, or mute a beat, all as ways to add rhythmic complexity to your strumming. And there is no rule that says you have to strum all the strings. Hit a single string instead of strumming them all. Play a series of single notes while playing a chord with your left hand and you will be basically playing an arpeggio. If you do that in the same rhythm you would use strumming, then you are really flatpicking, especially if you can do so while strill stressing the melody line.
Now that advice starts with the assumption that you are proficient strumming chords. But another approach would be to start with scales instead of chords and to develop a flatpicking proficiency that way. The simplest scale would be the D major scale beginning on the open bass string and ending on the third fret of the middle string. Play that scale in eighth notes, alternating your picking direction so that you go out-in or in-out consistently. Then you will want to work on other scales up and down the neck. (Now that I think about it, Stephen Seifert recently posted a scale exercise on the Dulcimer School and there is an older one on arpeggios by Aaron O'rourke).
And while playing scales will definitely help your technique, it also provides you with pieces of music you can fit into songs. "Arkansas Traveler" is a classic song that can be jazzed up with simple pieces of scales. Once you are proficient playing those scales and you see how a piece of one can fit into one song, you'll start to see those places in lots of songs.
Now for the disclaimer: I am not the dulcimer player you might think I am based on this post. I learned to flatpick playing the guitar. I worked on scales and arpeggions on the guitar, not the dulcimer. I do plan on taking the dulcimer more seriously and working on my technique--and perhaps learning Stephen Seifert's version of Black Mountain Rag--but I am not there yet.
I know a lot of people suggest not working on exercises, for they are not as fun as "just playing," but it appears you are at a point in your dulcimer playing where you want to develop a different technique, and that technique does demand some focused practice.
Good luck and happy picking!