Well, basic googling shows that yes, 1:3:6 resonances are possible, but looking at the articles they seem to mean the opposite way around (i.e. those numbers being frequencies rather than periods). That would mean, in terms of periods, 1:2:6. That is, the inner moons are an octave apart, while the outer moon is a perfect twelfth further out. Intuitively, this seems to make sense: harmonies greater than an octave seem rare in space, perhaps because the bodies would just be too far apart; in 1:3:6, the moon at the twelfth is pulling the octave moon further out, and it's already a long way from the gravity of the first. Whereas in 1:2:6 (in terms of periods), the inner two moons are close, and can both collaborate in keeping the twelfth in line.
[plus, I don't know about you, but the perfect twelfth doesn't actually sound that great. But maybe that's cultural conditioning.]
So, I don't know. I'd guess you could get the resonance you want by having the innermost moon significantly larger, or something? In the '1:3:6' resonance a paper I found discusses, btw (by which I think they mean 1:2:6 in your terms), apparently inclination of the moons increases rapidly to create a 3D system, not a simple planar alignment. Eccentricities also increase, and this causes tidal heating. In our system, Miranda is believed to have escaped from a 1:3 resonance, but to have ended up with a ridiculous inclination as a result.
However, I don't think any of that matters here. Because you've got an earth-size planet and three moon-size moons, so you can probably wave plausibility goodbye from the start. The moon is already about as large as the earth can handle - triple the size of the moon and I think you've got problems! But more significantly, you're into a big n-body problem there. Without a really massive planet, your moons are going to collapse into unpredictable chaotic orbits and at least some of them are going to go flying off.
For comparison here: the largest Jovian moon is twice the size of Luna. But Jupiter is over 300 times larger than Earth. You can get Jovian-style moon systems in the Jovian system because Jupiter's in the order of 30,000 times larger than even the largest of its moons; whereas your planet would only be around 100 times larger than your moons (30 times larger than all your moons together). And for reference: the Sun is around 1,000 times larger than Jupiter - but even that's enough to put the barycenter of the system outside the sun when the planet line up in certain ways. Your moons, if they could stay, which I don't think they can, would really be yanking your planet around, about a centre way outside itself! It wouldn't so much be a planet and moons, as a quaternary planet system!