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You cannot safely or properly wire a 4-wire L14-30 (which needs two hots, a neutral, and a separate ground) using your existing 10-3 cable without issues. Your current setup already has problems that should be fixed. Here's a clear breakdown.
Key Terms and L14-30 Pinout

L14-30 is a 30A, 125/250V twist-lock connector for 240V circuits that also provide neutral (for 120V loads or accessories). Pins are typically labeled:
X and Y (or brass screws): The two hot legs (240V between them).
W (silver screw): Neutral (white wire).
G: Equipment ground (green or bare).

Your 10-3 cable normally has black, red, white (insulated), plus a bare copper ground. This gives you only three current-carrying conductors plus ground.

Problems with Your Current Setup

Black and red as hots to the double-pole 30A breaker: Correct.
White wire connected to the appliance's green ground and to the neutral bar in the panel: This is incorrect and unsafe. You are using the neutral wire as a ground (or bonding them improperly). Neutral carries normal current; ground is only for faults. This can create shock hazards and violates code.

Proper Wiring for L14-30
You need four conductors total (two hots + dedicated neutral + ground). Your 10-3 doesn't provide that.
Best and recommended solution: Run new 10/4 cable (black, red, white, green/bare). Then wire as follows:

Black → One hot (X terminal on L14-30).
Red → Other hot (Y terminal).
White → Neutral (W terminal) — this goes back to the neutral bar in the panel.
Green or bare → Ground (G terminal) — this goes to the ground bar in the panel.

At the panel: Double-pole 30A breaker for the two hots. White to neutral bar, green/bare to ground bar.
If You Must Use Existing 10-3 (Not Recommended)
This only works if your appliance does NOT need neutral (pure 240V load with no 120V components). In that case:

Use black + red for the two hots (X and Y).
Use the bare/green as the ground to G on the plug.
Cap off and do not use the white wire (or repurpose it carefully if allowed, but this is non-standard).

You would then have no neutral at the plug, which means it won't fully match a standard L14-30 (designed for neutral). Adapters or modifications are possible for some generator or specific loads but introduce risks and may not be code-compliant.
Do not connect white to ground on the appliance or use it as ground.
Important Safety Notes

This is 240V/30A work — mistakes can cause fire, shock, or damage. If you're not experienced with electrical code and wiring, hire a licensed electrician.
Check your appliance manual for exact requirements (does it need 120V neutral?).
Local codes vary; permits may be needed for modifications.
Ensure the breaker is off and wires are de-energized before working.
Use proper strain relief, wire connectors, and torque specs on the plug terminals.

If the appliance is something like a generator inlet, welder, or RV, or if you can share the exact appliance model/make, I can give more tailored advice. Strongly consider upgrading to 10/4 cable for full compliance and safety.