Nothing in your description sounds wildly implausible. I'll just go through and extend some of your properties to make sure they make sense though.
The planet has three times Earth's mass
I'll assume that your new planet is Earth-like in composition and density. That implies that it's radius should be about $R_p \approx 1.4 \: R_\oplus$. Let's put that aside to use for later.
Now let's look at your moons.
- the first one to be as large as our moon to the sky
- the second 1.5 times larger
- and the third, half our moon's size.
To achieve this I have assumed:
- the first moon having 1.1 lunar radius,
- the second 2.3
- and the third 1.1.
The first moon, you want to be as large as our current Moon in the sky and you say it has a physical radius of $R_{M1} = 1.1\:R_{M}$. Our current moon subtends about $30\:arcsec = 8.73\times10^{-3}\:rad$ on average. You can calculate the distance, $d_{M1}$, the moon must be from the planet to have the same apparent size as our moon with the following equation.
$$d = \frac{R}{tan(\delta/2)}$$
Using $R_{M1}$ for $R$ and $8.73\times10^{-3}\:rad$ for $\delta$, your moon must be $d \approx 4\times10^8\:m = 44.5\:R_p$. Your first moon must be about 44.5 planetary radii out. Compare this to the Moon's distance of about 60 planetary radii out.
Now, we want to know how long it would take for such a moon to orbit, given the planet's mass, the moon's mass, and orbital distance. You can get that from Kepler's third law.
$$P = \sqrt{\frac{4\pi^2}{G(M_p + M_{M1})}d^3}$$
We can say $M_p = 3\:M_\oplus$ (as you specified) and we calculated $d$ just now. We only need to specify the mass of the moon. You provided the desired masses for your moons, so we're all set. Note $G = 6.67\times10^{-11} m^3kg^{-1}s^{-2}$ is the gravitational constant. I find that $P_{M1} = 1.44\times10^6\:s=16\:days$.
Let's stop right here for a second now. We've reached a point where your numbers are in conflict. Given your planet and your first moon's apparent size, mass, and radius, we found that it would have to orbit the planet in 16 days. If you want to keep the 1:2:4 resonance, then you need your other moons to orbit in 32 and 64 days, respectively.
I won't go through the rest of the math, I'll leave that up to you. But what you can do, if you want to ensure everything is consistent with real physics, is say I know how long my remaining two moons need to orbit for (i.e., you know $P$ for them), then at what distance must they orbit (i.e., what is the value of $d$ for them)? Work Kepler's third law backwards to get that. Then, given your desired apparent sizes in the sky, determine how large must they be physically by working the angular size equation backwards. You'll get new radii for the moon. You might then want to verify that the new mass and radii for your moons correspond to densities that match the desired moon types. E.g., if your third moon is going to be an ice moon, it should have a density of $\sim1\:g/cm^3$. You can play with your numbers until you get a system that matches your requires and fits the equations.
An important note: All these equations use MKS units. That is, masses should be in kilograms ($kg$), distances and sizes in meters ($m$), time in seconds ($s$), and angles in radians ($rad$).
