New update

You should not do a full loop during each switch, but only one iteration of the loop.

I did not run the sketch, just out of my head.

Some tips: print out the variables during each loop, so you get a clear idea what is happening (so all loop states and indices, and the periodIndex).

int LED = 3;  
int LED2 =10;
int LED3 = 9;

bool loop1state, loop2state, loop3state; // True if running, false if off
int  loop1index, loop2index, loop3index; // Iterator values
int  periodIndex = 0; // starting with 0, increasing every 10 ms


void setup()
{
  Serial.begin(9600);
}

void loop()
{
  // Update loop enablers.
  switch(periodIndex)
  {
  case 0:
    loop1state = true;
    loop1index = 0;
    break;

  case 1000:
    loop2state = true;
    loop2index = 0;
    break;

  case 1500:
    loop3state = true;
    loop3index = 0;
    break;

  case 4000:
     periodIndex = 0;
     break;
  }

  if (loop1state)
  {
    analogWrite(LED,loop1index++);
    loop1state = loop1index < 256;
  }

  if (loop2state)
  {
    analogWrite(LED2,loop2index++);
    loop2state = loop2index < 256;
  }

  if (loop3state)
  {
    analogWrite(LED3,loop3index++);
    loop3state = loop3index < 256;
  }

  periodIndex++;
  delay(10);
}

void loop()
{
  // Update loop enablers.
  switch(periodIndex)
  {
  case 0:
     loop1state = true;
     loop1index = 0;
     break;

  case 1000:
     loop2state = true;
     loop2index = 0;
     break;

  case 1500:
     loop3state = true;
     loop3index = 0;
     break;

  case 4000:
     periodIndex = 0;
     break;
  }

  if (loop1state)
  {
     DoLoop1(loop1index++);
     loop1state = loop1index < 256;
  }

  if (loop2state)
  {
     DoLoop2(loop2index++);
     loop2state = loop2index < 256;
  }

  if (loop3state)
  {
     DoLoop3(loop3index++);
     loop3state = loop3index < 256;
  }
}

void loop()
{
  // Update loop enablers.
  switch(periodIndex)
  {
  case 0:
     loop1state = true;
     loop1index = 0;
     break;

  case 1000:
     loop2state = true;
     loop2index = 0;
     break;

  case 1500:
     loop3state = true;
     loop3index = 0;
     break;

  case 4000:
     periodIndex = 0;
     break;
  }

  if (loop1state)
  {
     DoLoop1(loop1index++);
     loop1state = loop1index < 256;
  }

  if (loop2state)
  {
     DoLoop2(loop2index++);
     loop2state = loop2index < 256;
  }

  if (loop3state)
  {
     DoLoop3(loop3index++);
     loop3state = loop3index < 256;
  } 

  delay(10); // 10 ms
}

Update

Since you need all three loops together (and mixed), it's easier to define some variables.

bool loop1state, loop2state, loop3state; // True if running, false if off
int  loop1index, loop2index, loop3index; // Iterator values
int  periodIndex = 0; // starting with 0, increasing every 10 ms

Too bad I don't know the conditions when a loop starts, but let's say each iteration is 10 ms, and loop 1 starts immediately, loop 2 after 1000 ms, loop 3 after 1500 ms. After 3000 ms it starts all over.

void loop()
{
  // Update loop enablers.
  switch(periodIndex)
  {
  case 0:
     loop1state = true;
     loop1index = 0;
     break;

  case 1000:
     loop2state = true;
     loop2index = 0;
     break;

  case 1500:
     loop3state = true;
     loop3index = 0;
     break;

  case 4000:
     periodIndex = 0;
     break;
  }

  if (loop1state)
  {
     DoLoop1(loop1index++);
     loop1state = loop1index < 256;
  }

  if (loop2state)
  {
     DoLoop2(loop2index++);
     loop2state = loop2index < 256;
  }

  if (loop3state)
  {
     DoLoop3(loop3index++);
     loop3state = loop3index < 256;
  }
}

Note, not tested, but the idea shold be clear